Question

Calculate the resonant frequency and Q-factor (Quality factor) of a series L-C-R circuit containing a pure inductor of \[4\text{ H}\], capacitor of capacitance \[27\text{ }\mu \text{F}\] and resistor of resistance \[8.4\text{ }\Omega \].

Answer 1

Hint: At resonant frequency, the current in an L-C-R circuit will be maximum, which is possible only when inductive reactance \[{{X}_{L}}\] is equal to capacitive reactance \[{{X}_{c}}\].
At \[\omega ={{\omega }_{o}}\], \[{{X}_{L}}={{X}_{c}}\], that is,
\[{{\omega }_{o}}L=\dfrac{1}{{{\omega }_{o}}C}\]

Formula used:
The resonant frequency \[{{f}_{o}}\] of a series L-C-R is given by
\[{{f}_{o}}=\dfrac{1}{2\pi \sqrt{LC}}\]
Here L denotes the inductance of the inductor and C denotes the capacitance of the capacitor.
The Q-factor of the circuit is s given by
\[Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}\]
Here L denotes the inductance of the inductor, C denotes the capacitance of the capacitor, and R denotes the resistance of the resistor.

Complete step by step solution:
Inductance, \[L=4\text{ H}\]
Capacitance, \[C=27\text{ }\mu \text{F}\]
Resistance, \[R=8.4\text{ }\Omega \]
To find the resonant frequency, substitute the values of L and C in the resonant frequency formula:
\[\begin{align}
  & {{f}_{o}}=\dfrac{1}{2\pi \sqrt{(4\text{ H)(27 }\mu \text{F)}}} \\
 & {{f}_{o}}=\dfrac{1}{2\pi \sqrt{(4\text{ H)(27}\times \text{1}{{\text{0}}^{-6}}\text{F)}}} \\
 & {{f}_{o}}=15.31\text{ Hz} \\
\end{align}\]
Now, to find the Q-factor of the circuit, substitute the values of L, C, and R in the Q-factor frequency formula:
\[\begin{align}
  & Q=\dfrac{1}{8.4\text{ }\Omega }\sqrt{\dfrac{4\text{ H}}{27\text{ }\mu \text{F}}} \\
 & Q=\dfrac{1}{8.4\text{ }\Omega }\sqrt{\dfrac{4\text{ H}}{\text{27}\times \text{1}{{\text{0}}^{-6}}\text{F}}} \\
 & Q=45.82 \\
\end{align}\]

Therefore, the resonant frequency of the L-C-R circuit is \[15.31\text{ Hz}\] and the Q-factor is \[45.82\].

Additional information:
In general resonance is a phenomenon in which the natural frequency of a harmonic oscillator matches with the frequency of an external periodic force which in turn increases the amplitude of vibration.
Resonance is the result of oscillations in an L-C-R circuit as stored energy is passed from the inductor to the capacitor.
The sharpness of resonance is quantitatively described by a dimensionless number known as Q-factor or quality factor which is numerically equal to ratio of resonant frequency to bandwidth. The bandwidth is equal to L/R.

Note: The Q-factor of a series L-C-R circuit will be large if R is low, C is low or L is large.